Question

If \(\vec{n}_1\), \(\vec{n}_2\), and \(\vec{i}\) represent unit vectors along the incident ray, reflected ray, and normal to the surface, respectively, then:

• A. \(\vec{n}_2 = \vec{n}_1 - 2(\vec{n}_1 \cdot \hat{t})\hat{t}\)
• B. \(\vec{n}_2 = \vec{n}_1 + 2(\vec{n}_1 \cdot \hat{t})\hat{t}\)
• C. \(\vec{n}_2 = -\vec{n}_1\)
• D. \(\vec{n}_2 = 2\vec{n}_1 - (\vec{n}_1 \times \hat{t})\)

Hint:

The law of reflection states that the angle of incidence equals the angle of reflection. In vector form, this can be represented using the normal vector andthe dot product for the direction of the reflected ray.

Answer 1

The Correct Option is B

1. Step 1: According to the laws of reflection, the incident ray, the reflected ray, and the normal to the surface all lie in the same plane, and the angle of incidence equals the angle of reflection.
2. Step 2: Using the vector representation of reflection, we have:\[ \vec{n}_2 = \vec{n}_1 + 2(\vec{n}_1 \cdot \hat{t})\hat{t} \]
This is the correct vector equation for the reflected ray direction based on the law of reflection.

Answer 2

The hint provides the following unit vectors in a coordinate system where $\hat{t}$ is the normal to the surface and $\hat{i}$ is tangential to the surface in the plane of incidence:

• Incident ray unit vector: $\hat{n}_1 = -\cos\theta \hat{t} + \sin\theta \hat{i}$
• Reflected ray unit vector: $\hat{n}_2 = \cos\theta \hat{t} + \sin\theta \hat{i}$

From these, the hint also states:

$\hat{n}_2 - \hat{n}_1 = 2\cos\theta \hat{t} = -2(\hat{n}_1 \cdot \hat{t}) \hat{t}$

and

$\hat{n}_1 \cdot \hat{t} = -\cos\theta$

The final boxed result from the hint is the vector form of the law of reflection:

$\hat{n}_2 = \hat{n}_1 - 2(\hat{n}_1 \cdot \hat{t}) \hat{t}$

Explanation:

• The expressions for $\hat{n}_1$ and $\hat{n}_2$ explicitly show the law of reflection: the angle of incidence ($\theta$) is equal to the angle of reflection ($\theta$), and the tangential component ($\sin\theta \hat{i}$) remains unchanged. The normal component reverses direction ($-\cos\theta \hat{t}$ becomes $\cos\theta \hat{t}$).
• The dot product $\hat{n}_1 \cdot \hat{t} = -\cos\theta$ captures the cosine of the angle between the incident ray and the normal.
• The vector formula $\hat{n}_2 = \hat{n}_1 - 2(\hat{n}_1 \cdot \hat{t}) \hat{t}$ provides a general way to find the reflected ray unit vector given the incident ray unit vector and the normal unit vector, without explicitly needing to define a tangential unit vector. The term $2(\hat{n}_1 \cdot \hat{t}) \hat{t}$ represents twice the projection of $\hat{n}_1$ onto $\hat{t}$, and subtracting it from $\hat{n}_1$ effectively reverses the component of $\hat{n}_1$ along the normal.